Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#48836 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C22×#562 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7026 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 38 points:
(1,2);(4,6)(7,8,12,14)(9,15,16,10)(11,13,19,20)(17,21,22,18)(24,28)(26,34)(29,31)(32,38);(9,16)(10,15)(11,19)(13,20)(23,24,27,28)(25,29,33,31)(26,35,34,30)(32,37,38,36);(3,4,5,6)(7,9,12,16)(8,10,14,15)(11,17,19,22)(13,18,20,21)(23,24)(25,31)(26,30)(27,28)(29,33)(32,37)(34,35)(36,38);(7,10)(8,9)(11,18)(12,15)(13,17)(14,16)(19,21)(20,22)(23,25,27,33)(24,29,28,31)(26,32,34,38)(30,36,35,37);(7,11)(8,13)(9,17)(10,18)(12,19)(14,20)(15,21)(16,22)(23,26,27,34)(24,30,28,35)(25,32,33,38)(29,36,31,37)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7612125);